Tracking Paths

  • Aritra Banik
  • Matthew J. Katz
  • Eli Packer
  • Marina Simakov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)


We consider several problems dealing with tracking of moving objects (e.g., vehicles) in networks. Given a graph \(G=(V,E)\) and two vertices \(s,t \in V\), a set of vertices \(T \subseteq V\) is a tracking set for G (w.r.t. paths from s to t), if one can distinguish between any two paths from s to t by the order in which the vertices of T appear (or do not appear) in them. We prove that the problem of finding a minimum-cardinality tracking set w.r.t. shortest paths from s to t is NP-hard and even APX-hard. On the other hand, for the common case where G is planar, we present a 2-approximation algorithm for this problem. We also consider the following related problem: Given a graph G, two vertices s and t, and a set of forbidden vertices \(V_F \subseteq V-\{s,t\}\), find a minimum-cardinality set of trackers \(V^* \subset V\), such that a shortest path P from s to t passes through a forbidden vertex if and only if it passes through a vertex of \(V^*\). We present a polynomial-time (exact) algorithm for this problem.


  1. 1.
    Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discret. Math. 12(3), 289–297 (1999)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bar-Yehuda, R., Geiger, D., Naor, J.S., Roth, R.M.: Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and bayesian inference. SIAM J. Comput. 27, 942–959 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bhatti, S., Xu, J.: Survey of target tracking protocols using wireless sensor network. In: Proceedings of 5th International Conference on Wireless and Mobile Communications, pp. 110–115 (2009)Google Scholar
  4. 4.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry Algorithms and Applications, 3rd edn. Springer, Heidelberg (2008)MATHGoogle Scholar
  5. 5.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162, 439–485 (2005)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ganesan, D., Cristescu, R., Beferull-Lozano, B.: Power-efficient sensor placement and transmission structure for data gathering under distortion constraints. ACM Trans. Sens. Netw. 2(2), 155–181 (2006)CrossRefGoogle Scholar
  7. 7.
    Karger, D.R., Stein, C.: A new approach to the minimum cut problem. J. ACM 43(4), 601–640 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Karp, R.M.: Reducibility among combinatorial problems. In: Proceedings of a Symposium on the Complexity of Computer Computations, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, pp. 85–103. Plenum Press (1972)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Aritra Banik
    • 1
  • Matthew J. Katz
    • 2
  • Eli Packer
    • 3
  • Marina Simakov
    • 2
  1. 1.Indian Institute of TechnologyJodhpurIndia
  2. 2.Ben-Gurion University of the NegevBeer-ShevaIsrael
  3. 3.Haifa LabIBM ResearchHaifaIsrael

Personalised recommendations